Circumcenter is a classical concept in geometry. Recently, circumcenters have been used to accelerate iterative methods, such as the Douglas-Rachford method and method of alternating projections, in optimization; see e.g., [1]}, [2]}, [3]}, [4]}, [5]}. Compared with the classic Douglas-Rachford method and the method of alternating projections, the circumcentered methods present much better performance for solving the best approximation problems and feasibility problems.
In [6]}, we provided a systematic study on the circumcenter of a finite set in a Hilbert space.
This allowed us to investigate circumcentered
methods of nonexpansive mappings, isometries, and best approximation mappings; see, e.g., [7]}, [8]}, [9]}.
Bregman distances have also been widely studied in optimization;
see [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]} and references therein.
A natural question is: Can one define circucmcenters using Bregman distances and use them to accelerate
iterative methods? However, up to now a study of circumcenters in
the framework of Bregman distances is still missing in the literature.
